Recent years have witnessed increasing interest in optimization proxies, i.e., machine learning models that approximate the input-output mapping of parametric optimization problems and return near-optimal feasible solutions. Following recent work by (Nellikkath & Chatzivasileiadis, 2021), this paper reconsiders the optimality verification problem for optimization proxies, i.e., the determination of the worst-case optimality gap over the instance distribution. The paper proposes a compact formulation for optimality verification and a gradient-based primal heuristic that brings substantial computational benefits to the original formulation. The compact formulation is also more general and applies to non-convex optimization problems. The benefits of the compact formulation are demonstrated on large-scale DC Optimal Power Flow and knapsack problems.

Variational Autoencoders for multimodal data hold promise for many tasks in data analysis, such as representation learning, conditional generation, and imputation. Current architectures either share the encoder output, decoder input, or both across modalities to learn a shared representation. Such architectures impose hard constraints on the model. In this work, we show that a better latent representation can be obtained by replacing these hard constraints with a soft constraint. We propose a new mixture-of-experts prior, softly guiding each modality's latent representation towards a shared aggregate posterior. This approach results in a superior latent representation and allows each encoding to preserve information better from its uncompressed original features. In extensive experiments on multiple benchmark datasets and two challenging real-world datasets, we show improved learned latent representations and imputation of missing data modalities compared to existing methods.

We study the addition of shape constraints and their consideration during the parameter estimation step of symbolic regression (SR). Shape constraints serve as a means to introduce prior knowledge about the shape of the otherwise unknown model function into SR. Unlike previous works that have explored shape constraints in SR, we propose minimizing shape constraint violations during parameter estimation using gradient-based numerical optimization. We test three algorithm variants to evaluate their performance in identifying three symbolic expressions from a synthetically generated data set. This paper examines two benchmark scenarios: one with varying noise levels and another with reduced amounts of training data. The results indicate that incorporating shape constraints into the expression search is particularly beneficial when data is scarce. Compared to using shape constraints only in the selection process, our approach of minimizing violations during parameter estimation shows a statistically significant benefit in some of our test cases, without being significantly worse in any instance.

Modern power systems integrate renewable distributed energy resources (DERs) as an environment-friendly enhancement to meet the ever-increasing demands. However, the inherent unreliability of renewable energy renders developing DER management algorithms imperative. We study the energy-sharing problem in a system consisting of several DERs. Each agent harvests and distributes renewable energy in its neighborhood to optimize the network's performance while minimizing energy waste. We model this problem as a bandit convex optimization problem with constraints that correspond to each node's limitations for energy production. We propose distributed decision-making policies to solve the formulated problem, where we utilize the notion of dynamic regret as the performance metric. We also include an adjustment strategy in our developed algorithm to reduce the constraint violations. Besides, we design a policy that deals with the non-stationary environment. Theoretical analysis shows the effectiveness of our proposed algorithm. Numerical experiments using a real-world dataset show superior performance of our proposal compared to state-of-the-art methods.

Geometry is a ubiquitous language of computer graphics, design, and engineering. However, the lack of large shape datasets limits the application of state-of-the-art supervised learning methods and motivates the exploration of alternative learning strategies. To this end, we introduce geometry-informed neural networks (GINNs) to train shape generative models \emph{without any data}. GINNs combine (i) learning under constraints, (ii) neural fields as a suitable representation, and (iii) generating diverse solutions to under-determined problems. We apply GINNs to several two and three-dimensional problems of increasing levels of complexity. Our results demonstrate the feasibility of training shape generative models in a data-free setting. This new paradigm opens several exciting research directions, expanding the application of generative models into domains where data is sparse.

We study the problem of online convex optimization (OCO) under unknown linear constraints that are either static, or stochastically time-varying. For this problem, we introduce an algorithm that we term Optimistically Safe OCO (OSOCO) and show that it enjoys $\tilde{\mathcal{O}}(\sqrt{T})$ regret and no constraint violation. In the case of static linear constraints, this improves on the previous best known $\tilde{\mathcal{O}}(T^{2/3})$ regret with only slightly stronger assumptions. In the case of stochastic time-varying constraints, our work supplements existing results that show $\mathcal{O}(\sqrt{T})$ regret and $\mathcal{O}(\sqrt{T})$ cumulative violation under more general convex constraints albeit a less general feedback model. In addition to our theoretical guarantees, we also give numerical results comparing the performance of OSOCO to existing algorithms.

Meta-reinforcement learning has widely been used as a learning-to-learn framework to solve unseen tasks with limited experience. However, the aspect of constraint violations has not been adequately addressed in the existing works, making their application restricted in real-world settings. In this paper, we study the problem of meta-safe reinforcement learning (Meta-SRL) through the CMDP-within-online framework to establish the first provable guarantees in this important setting. We obtain task-averaged regret bounds for the reward maximization (optimality gap) and constraint violations using gradient-based meta-learning and show that the task-averaged optimality gap and constraint satisfaction improve with task-similarity in a static environment or task-relatedness in a dynamic environment. Several technical challenges arise when making this framework practical. To this end, we propose a meta-algorithm that performs inexact online learning on the upper bounds of within-task optimality gap and constraint violations estimated by off-policy stationary distribution corrections. Furthermore, we enable the learning rates to be adapted for every task and extend our approach to settings with a competing dynamically changing oracle. Finally, experiments are conducted to demonstrate the effectiveness of our approach.

Motivated by the need to solve open-loop optimal control problems with computational efficiency and reliable constraint satisfaction, we introduce a general framework that combines diffusion models and numerical optimization solvers. Optimal control problems are rarely solvable in closed form, hence they are often transcribed into numerical trajectory optimization problems, which then require initial guesses. These initial guesses are supplied in our framework by diffusion models. To mitigate the effect of samples that violate the problem constraints, we develop a novel constrained diffusion model to approximate the true distribution of locally optimal solutions with an additional constraint violation loss in training. To further enhance the robustness, the diffusion samples as initial guesses are fed to the numerical solver to refine and derive final optimal (and hence feasible) solutions. Experimental evaluations on three tasks verify the improved constraint satisfaction and computational efficiency with 4$\times$ to 30$\times$ acceleration using our proposed framework, which generalizes across trajectory optimization problems and scales well with problem complexity.

In numerous reinforcement learning (RL) problems involving safety-critical systems, a key challenge lies in balancing multiple objectives while simultaneously meeting all stringent safety constraints. To tackle this issue, we propose a primal-based framework that orchestrates policy optimization between multi-objective learning and constraint adherence. Our method employs a novel natural policy gradient manipulation method to optimize multiple RL objectives and overcome conflicting gradients between different tasks, since the simple weighted average gradient direction may not be beneficial for specific tasks' performance due to misaligned gradients of different task objectives. When there is a violation of a hard constraint, our algorithm steps in to rectify the policy to minimize this violation. We establish theoretical convergence and constraint violation guarantees in a tabular setting. Empirically, our proposed method also outperforms prior state-of-the-art methods on challenging safe multi-objective reinforcement learning tasks.

Lane detection is a fundamental task in autonomous driving. While the problem is typically formulated as the detection of continuous boundaries, we study the problem of detecting lane boundaries that are sparsely marked by 2D points with many false positives. This problem arises in the Formula Student Driverless (FSD) competition and is challenging due to its inherent ambiguity. Previous methods are inefficient and unable to find long-horizon solutions. We propose a deterministic algorithm called CLC that uses backtracking graph search with a learned likelihood function to overcome these limitations. We impose geometric constraints on the lane candidates to guarantee a geometrically sound lane. Our exhaustive search leads to finding the global optimum in 45% of instances, and the algorithm is overall robust to up to 50% false positives. Our algorithm runs in less than 15 ms on a single CPU core, meeting the low latency requirements of autonomous racing. We extensively evaluate our method on real data and realistic racetrack layouts, and show that it outperforms the state-of-the-art by detecting long lanes over 100 m with few (0.6%) critical failures. This allows our autonomous racecar to drive close to its physical limits on a previously unknown racetrack without being limited by perception. We release our dataset with realistic Formula Student racetracks to enable further research.